姓名：温广辉 电话：010-62742440 单位：北京大学工学院力学与空天技术系 传真：010-62765037 力学系统与控制专业 2008 级博士生 邮件：wenguanghui@gmail.com

第六届全国复杂网络学术会议最佳学生论文奖申请理由

评审老师，您好： 本文是在访问陈关荣教授时完成，虞文武博士，苏厚胜老师，严钢博士也给

予了很多指导，使文章润色不少，在此致谢。 多智能体系统一致性控制受到广泛关注，有着深刻理论意义和实际应用价值

的结果春笋般涌现。目前，针对连续多智能体系统，大部分结果均基于连续信息

传输这一假设而建立；实际中，该假设难以成立。譬如，声纳障碍导致信息传输

丢包现象的发生，智能体之间信息传输呈现间歇性。因此，本文考虑具有间歇状

态测度的一致性问题。此外，实际多智能体系统都是非线性的，线性只是非线性

的一种近似。进一步地，本文研究了具有时滞非线性动力学行为和间歇信息测度

的二阶多智能体系统的一致性问题。结果表明，当通讯时间大于某一阈值时，多

智能体系统一致性的实现对于间歇通讯具有鲁棒性，并给出了该阈值依赖于网络

拓扑特性和系统动力学特性的解析表达式，具有一定的理论价值和实际应用价

值。 因此，向大会提交本文并申请第六届复杂网络学术会议最佳学生论文奖。

此致 敬礼！ 温广辉

2010 年 8 月 8 日

个人在读证明（学生证扫描）

Second-order consensus of multi-agent systems with delayed nonlinear dynamicsand intermittent measurements

Guanghui Wen and Zhisheng Duan

State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, P. R. China

Abstract

This paper investigates second-order consensus of multi-agent systems with intrinsic delayed nonlinear dynamics

and switching topologies. Each agent is assumed to obtain the measurements of relative states between its own

and the neighbors only at a sequence of disconnected time intervals. A novel intermittent consensus protocol is

proposed to guarantee the states of agents with time-varying velocities to reach second-order consensus under a

fixed strongly connected and balanced topology. The results are then extended to second-order consensus in multi-

agent systems with switching topologies, where each possible communication topology is strongly connected and

balanced. By virtue of a Lyapunov control approach, it is shown that consensus can be reached if the general algebraic

connectivity and communication time duration are larger than their corresponding threshold values, respectively.

Finally, simulation examples are provided to verify theoretical analysis and the effectiveness of the new protocol.

Keywords: Multi-agent system, second-order consensus, intermittent measurement, delayed nonlinear dynamics.

1. Introduction

Recently, cooperative control has received considerable attention partly owing to its wide applications in multi-

agents systems, where typical examples include state consensus seeking of multiple mobile vehicles [1, 2, 3], design

of distributed sensor networks [4], control of flocking and rendezvous in natural as well as social systems [5, 6, 7].

Among the numerous research topics in cooperative control, consensus problem received particular interests [8, 9, 10],

which can be generally described as how to design an appropriate protocol based on the local information under some

communication topologies to ensure the multiple agents to reach an agreement on certain quantities of interest.

In Ref. [5], Vicsek et al. introduced an interesting discrete-time model of mobile agents, where each agent’s

motion is updated according to a local rule based on its own state as well as the states of its neighbors. Some the-

oretical analysis of the consensus problem on the linearized Vicsek’s mode was provided in Ref. [8]. Then, in Ref.

[9], a general framework of the consensus problem for networks of dynamic agents with fixed or switching topology

and communication time-delays was established. The consensus conditions derived in Ref. [9] were further relaxed

in Ref. [10]. In addition, consensus over a random communication topology [11, 12], asynchronous consensus

[13, 14, 15], high-dimensional consensus [16], consensus problems with nonlinear protocols [17, 18] and commu-

nication noises [19, 20], have been investigated. Note that most of the above-mentioned work are concerned with

the first-order consensus problem, where each agent is governed by first-order dynamics. In reality, however, a large

class of multi-agent systems are modeled by second-order dynamics [21], [22], [23] [24], [25], [26]. In Ref. [21],

second-order consensus problem with zero initial and finial consensus velocities under undirected communication

Preprint submitted to CCCN–2010 August 15, 2010

topology was investigated. Taking into account the general case where information flow may be unidirectional due to

sensors with limited sensing ranges or multi-agents with directed communication links, a new kind of second-order

consensus problems under directed communication topology was discussed in Refs. [22, 23]. Some sufficient condi-

tions were obtained for achieving second-order consensus, and it was shown that the communication topology having

a spanning tree is not a sufficient condition for reaching second-order consensus, which is different in kind from the

first-order consensus problems [22, 23, 24]. Then, in Refs. [27, 28, 29], some necessary and sufficient conditions for

second-order consensus in directed networks were derived. Concerning that transmission time-delay is a key factor

influencing the stability of consensus in linear multi-agent systems, a necessary and sufficient condition was obtained

for second-order consensus in networked multi-agent systems with transmission delays in Ref. [29]. In contrast to

the aforementioned second-order consensus algorithms [21], [22], [24] [27], [28], where the consensus velocity is a

constant, a consensus algorithm in coupled second-order linear harmonic oscillators with asymptotic periodic veloc-

ity and directed communication topology was considered in Ref. [30]. The dynamical model studied in Ref. [30] in

essence is a second-order multi-agent system with intrinsic linear dynamics. A more general case is that each agent

has its intrinsic nonlinear dynamics [31, 32, 33, 34, 35, 36]. From this perspective, Yu et al. investigated the second-

order consensus problem in multi-agent systems with nonlinear dynamics and directed topologies in Ref. [37], where

by using tools from algebraic graph theory and Lyapunov control approach, some sufficient conditions were derived

for reaching second-order consensus with time-varying consensus velocities.

It should be noticed that most of the aforementioned works on second-order consensus problems in multi-agent

systems, it was assumed that information is transmitted continuously among multi-agents. However, this may not be

the case in reality due to technological limitations or external disturbances. For example, in some cases, agents can

only obtain the measurements of states of its neighbors intermittently due to the limited of sensing abilities. To deal

with this challenging situation, a novel intermittent consensus protocol is proposed in this paper to guarantee second-

order consensus. On the other hand, in order to analyze the second-order consensus problem in multi-agent systems

within a general framework, intrinsic delayed nonlinear dynamics are introduced into the model of each agent in this

paper. By virtue of the Lyapunov control approach, some sufficient conditions are derived for reaching second-order

consensus with time-varying agent velocities.

The rest of the paper is organized as follows. In Section 2, some preliminaries in algebraic graph theory and the

model formulation are given. In Section 3, second-order consensus problems with delayed nonlinear dynamics and

intermittent measurements under fixed and switching strongly connected and balanced communication topologies are

investigated, respectively. In Section 4, simulation examples are provided to verify the theoretical results. Conclusions

are finally drawn in Section 5.

The following notations are used throughout the paper. Let ℝ and ℕ be the sets of real and natural numbers,

respectively. ℝN is the N-dimensional real vector space and ∥ · ∥ denotes the Euclidian norm. ℝN×N is N × N real

matrix space. Let IN (ON) be the N−dimensional identity (zero) matrix, and 1N (0N) be the N−dimensional column

vector with each entry being 1 (0). Suppose that matrix M ∈ ℝN×N has real eigenvalues, with λi(M) being the ith

smallest eigenvalue (1 ≤ i ≤ N). Notation ⊗ represents the Kronecker product. Furthermore, a column vector x ∈ ℝN

is said to be positive if every entry xi > 0 (1 ≤ i ≤ N).

2

2. Preliminaries

In this section, some preliminaries in algebraic graph theory and model formulation for second-order consensus

in multi-agent systems with delayed nonlinear dynamics and intermittent measurements are introduced.

2.1. Algebraic graph theory

LetG(V,E,A) be a directed graph with the set of nodesV = {v1, v2, · · · , vN}, the set of directed edges E ⊆ V×V,

and a weighted adjacency matrix A = [ai j]N×N with non-negative adjacency elements ai j. An edge ei j in graph G is

denoted by the ordered pair of nodes (v j, vi), where v j and vi are called the parent and child nodes, respectively, and

ei j ∈ E if and only if ai j > 0. Furthermore, self-loops are not allowed, i.e., aii = 0 for all i = 1, 2, · · ·N. For simplicity,

denote G(V,E,A) by G(A) if no confusion will arise.

A directed path from node vi to v j is a finite ordered sequence of edges, (vi, vk1 ), (vk1 , vk2 ), · · · , (vkl , v j), with

distinct nodes vkm , m = 1, 2, · · · , l. A directed graph is called strongly connected if and only if there is a directed path

between any pair of distinct nodes. Moreover, a directed graph G(A) is called balanced if∑j

ai j =∑

j

a ji, ∀ i = 1, 2, · · · ,N. (1)

The Laplacian matrix L = [li j]N×N of G(A) is defined as

li j =

− ai j, i , j,

N∑k=1,k,i

aik, i = j.(2)

For a directed graph, the Laplacian matrix L has the following properties.

Lemma 1: ([10]) Suppose that a directed graph G(A) is strongly connected. Then, 0 is a simple eigenvalue of its

Laplacian matrix L, and all the other eigenvalues of L have positive real parts.

Lemma 2: ([9]) A directed graph G(A) is balanced if and only if 1N is the left eigenvector of its Laplacian matrix

L associated with zero eigenvalue, i.e. 1TN L = 0.

For an undirected graph, its Laplacian matrix L is positive semi-definite. For a connected undirected graph, there

is one simple zero eigenvalue of L, and all the other eigenvalues of L are positive and real.

2.2. Formulation of the model

Consider a group of N agents indexed by 1, 2, · · · ,N. The commonly studied continuous-time second-order

protocol of the N agents is described as follows [23, 24, 26]:xi(t) = vi(t)

vi(t) = −αN∑

j=1

li jx j(t) − βN∑

j=1

li jv j(t), i = 1, 2, · · · ,N,(3)

where xi ∈ ℝn and vi ∈ ℝn are the position and velocity states of the ith agent, respectively, α and β represent the

coupling strengths, L = [li j]N×N is the Laplacian matrix of the fixed communication topology G(A). When the agents

reach second-order consensus, the velocities of all agents converge to∑N

j=1 ξ jv j(0), which depends only on the initial

3

velocities of the agents, where ξ = (ξ1, · · · , ξN) is the nonnegative left eigenvector of L associated with the eigenvalue

0 satisfying ξT 1N = 1 [23, 24]. However, in most applications of multi-agent formations, the velocity of each agent

is generally evolving nonlinearly. Therefore, Yu et al. proposed the following second-order consensus protocol with

nonlinear dynamics [37]xi(t) = vi(t)

vi(t) = f (xi(t), vi(t), t) − αN∑

j=1

li jx j(t) − βN∑

j=1

li jv j(t), i = 1, 2, · · · ,N,(4)

where f : ℝn × ℝn × ℝ+ → ℝn is a continuously differentiable vector-valued function. In some cases, f can be

taken as f = −▽U(x, v), where U(x, v) is a potential function, thus the multi-agent system (4) includes many popular

swarming and flocking models [38], [39] as special cases.

Note that most of the existing protocols are implemented based on a common assumption that all information is

transmitted continuously among agents. However, in some real situations, agents may only communicate with their

neighbors over some disconnected time intervals due to the unreliability of communication channels, failure of phys-

ical devices, and limitations of sensing ranges, etc. Motivated by this observation and based on the above-mentioned

works [23, 24, 37], in this paper the following consensus protocol with time-delay and intermittent measurements is

considered:

xi(t) = vi(t)

vi(t) = f (xi(t − τ), vi(t − τ), xi(t), vi(t), t) − αN∑

j=1

li jx j(t) − βN∑

j=1

li jv j(t), t ∈ [kω, kω + δ],

vi(t) = f (xi(t − τ), vi(t − τ), xi(t), vi(t), t), t ∈ (kω + δ, (k + 1)ω), k ∈ ℕ, i = 1, 2, · · · ,N,

(5)

where f : ℝn × ℝn × ℝn × ℝn × ℝ+ → ℝn is a continuously differentiable vector-valued function representing the

intrinsic delayed nonlinear dynamics of agent i, τ > 0 is the time-delay constant, and the communication time duration

δ satisfies τ < δ ≤ ω. Furthermore, xi(t) = ϕi(t), vi(t) = ψi(t), for t ∈ [−τ, 0], i = 1, 2, · · · , and the initial functions ϕi

and ψi are continuous for t ∈ [−τ, 0].

Clearly, since∑N

j=1 li j = 0, if consensus can be achieved, it is natural to require a solution s(t) = (sT1 (t), sT

2 (t))T ∈ℝ2n of the system (5) be a possible trajectory of an isolated node satisfying s1(t) = s2(t),

s2(t) = f (s1(t − τ), s2(t − τ), s1(t), s2(t), t).(6)

Here, s(t) may be an isolated equilibrium point, a periodic orbit, or even a chaotic orbit in applications.

Remark 1: If τ = 0 and δ = ω in system (5), that is, each agent can communicate with its neighbors all the time

and the node dynamics depend only on its current states, then system (5) becomes the system (3) studied in Ref. [37].

Lemma 3: (Schur complement [40]) The following linear matrix inequality (LMI),

S =

S 11 S 12

S 21 S 22

> 0,

where S 11 = S T11, S 12 = S T

21, S 22 = S T22, is equivalent to one of the following conditions:

4

(i) S 11 > 0, S 22 − S 21S −111 S 12 > 0;

(ii) S 22 > 0, S 11 − S 12S −122 S 21 > 0.

Lemma 4: (Halanay Inequality [41]) Suppose that the non-negative function y(t), t ∈ [−τ,+∞), satisfies

dy(t)dt≤ −c1y(t) + c2y(t − τ), t ≥ 0,

where constants c1 > c2 > 0. Then,

y(t) ≤ |y(0)|τe−rt, t ≥ 0,

where |y(0)|τ = max−τ≤s≤0

y(s) and r is the unique solution of

−r = −c1 + c2erτ.

Lemma 5: [42] Suppose that the non-negative function y(t), t ∈ [−τ,∞), satisfies

dy(t)dt≤ c1y(t) + c2y(t − τ), t ≥ 0,

where c1, c2 are positive constants. Then,

y(t) ≤ |y(0)|τe(c1+c2)t, t ≥ 0,

where |y(0)|τ = max−τ≤s≤0

y(s).

3. Main Results

In this section, second-order consensus problems in strongly connected and balanced networks with time-delayed

nonlinear dynamics and intermittent measurements are investigated.

Assumption 1: There exist nonnegative constants ρi, i ∈ {1, 2, 3, 4}, such that

∥ f (x1, x2, x3, x4, t) − f (y1, y2, y3, y4, t)∥ ≤4∑

i=1

ρi∥xi − yi∥,

∀xi, yi ∈ ℝn, i ∈ {1, 2, 3, 4}, t ≥ 0.

Let xi(t) = xi(t) − 1N

∑Nj=1 x j(t) and vi(t) = vi(t) − 1

N∑N

j=1 v j(t). One has the following error dynamical system:

˙xi(t) = vi(t),

˙vi(t) = f (xi(t − τ), vi(t − τ), xi(t), vi(t), t) −1N

N∑j=1

f (x j(t − τ), v j(t − τ), x j(t), v j(t), t)

− αN∑

j=1

li j x j(t) − βN∑

j=1

li jv j(t), kω ≤ t ≤ kω + δ,

˙vi(t) = f (xi(t − τ), vi(t − τ), xi(t), vi(t), t) −1N

N∑j=1

f (x j(t − τ), v j(t − τ), x j(t), v j(t), t),

kω + δ < t < (k + 1)ω, i = 1, · · · ,N, k = 0, 1, · · · .

(7)

5

Let x(t) = (xT1 (t), · · · , xT

N(t))T , v(t) = (vT1 (t), · · · , vT

N(t))T , f (x(t − τ), v(t − τ), x(t), v(t), t) = ( f T (x1(t − τ), v1(t −τ), x1(t), v1(t), t), · · · , f T (xN(t − τ), vN(t − τ), xN(t), vN(t), t))T and y(t) = (xT (t), vT (t))T . Then, system (7) can be

written as ˙y(t) = F(x(t − τ), v(t − τ), x(t), v(t), t) + (B1 ⊗ In)y(t), t ∈ [kω, kω + δ],

˙y(t) = F(x(t − τ), v(t − τ), x(t), v(t), t) + (B2 ⊗ In)y(t), t ∈ (kω + δ, (k + 1)ω),(8)

where F(x(t−τ), v(t−τ), x(t), v(t), t) =

0Nn[(IN − 1

N 1N×N) ⊗ In

]f (x(t − τ), v(t − τ), x(t), v(t), t)

, B1 =

ON IN

−αL −βL

,B2 =

ON IN

ON ON

.Theorem 1. Suppose that the communication topologyG(A) is strongly connected and balanced, and Assumption

1 holds. Then, second-order consensus in system (5) is achieved if the following conditions hold:

(i) λ2(L + LT ) > αβ2 ,

(ii) λ1(R1) > c0λ2(P1)λ1(Q) ,

(iii) δ > rτ+(γ3+γ4)ωr+γ3+γ4

,

where R1 =

(αλ2(L + LT ) − ρ1 − ρ2 − 2ρ3

)α − (βρ3 + αρ4)

− (βρ3 + αρ4) β2λ2(L + LT ) − (ρ1 + ρ2 + 2ρ4)β − 2α

, P1 =

αβλN(L + LT ) α

α β

,Q =

αβλ2(L + LT ) α

α β

, c0 = (α + β) max{ρ1, ρ2}, r is the unique positive solution of −r = −γ1 + γ2erτ, γ1 =

λ1(R1)λ2(P1) , γ2 =

c0λ1(Q) , γ3 =

c1+c2+√

(c1−c2)2+c3

λ1(Q) , γ4 =(α+β)ρ0λ1(Q) , c1 =

(ρ1+ρ2+2ρ3)α2 , c2 =

(ρ1+ρ2+2ρ4)β2 , and c3 = (βρ3+αρ4+1)2.

Proof: Construct the following Lyapunov function candidate

V(t) =12

yT (t)(P ⊗ In)y(t), (9)

where P =

αβ(L + LT ) αIN

αIN βIN

. It will be shown that V(t) is a valid Lyapunov function for analyzing the error

dynamics described by system (8). According to the Courant-Fischer theorem [43], one has

V(t) =αβ

2xT (t)

((L + LT ) ⊗ In

)x(t) + αxT (t)v(t) +

β

2vT (t)v(t)

≥ 12

yT (t)(Q ⊗ INn)y(t),

where Q =

αβλ2(L + LT ) α

α β

. By Lemma 3, Q > 0 is equivalent to both β > 0 and λ2(L + LT ) > αβ2 . From

condition (i), one obtains Q > 0, V(t) ≥ 0 and V(t) = 0 if and only if y(t) = 02Nn.

Let x(t − τ) = 1N

∑Nj=1 x j(t − τ), v(t − τ) = 1

N∑N

j=1 v j(t − τ), x(t) = 1N

∑Nj=1 x j(t), and v(t) = 1

N∑N

j=1 v j(t). For

6

t ∈ [kω, kω + δ], k ∈ ℕ, taking the time derivative of V(t) along the trajectories of (8) gives

V(t) = yT (t)(P ⊗ In)[F(x(t − τ), v(t − τ), x(t), v(t), t) + (B1 ⊗ In)y(t)]

= αxT[(

IN −1N

1N×N

)⊗ In

]f (x(t − τ), v(t − τ), x(t), v(t), t) + βvT

[(IN −

1N

1N×N

)⊗ In

]f (x(t − τ), v(t − τ), x(t), v(t), t)

+12

yT (t)[(

PB1 + BT1 P

)⊗ In

]y(t)

=[αxT (t) + βvT (t)

] [f (x(t − τ), v(t − τ), x(t), v(t), t) − 1N ⊗ f (x(t − τ), v(t − τ), x(t), v(t), t)

]−

[αxT (t) + βvT (t)

] (( 1N

1N×N

)⊗ In

)f (x(t − τ), v(t − τ), x(t), v(t), t)

+[αxT (t) + βvT (t)

] [1N ⊗ f (x(t − τ), v(t − τ), x(t), v(t), t)

]+

12

yT (t)

−α2(L + LT ) ON

ON −β2(L + LT ) + 2αIN

⊗ In

y(t) (10)

Since x(t) = [(IN − 1N 1N×N) ⊗ In]x(t) and v(t) = [(IN − 1

N 1N×N) ⊗ In]v(t), one gets

xT (t)[1N ⊗ f (x(t − τ), v(t − τ), x(t), v(t), t)

]= 0,

vT (t)[1N ⊗ f (x(t − τ), v(t − τ), x(t), v(t), t)

]= 0,

(11)

and

xT (t)[(

1N

1N×N

)⊗ In

]f (x(t − τ), v(t − τ), x(t), v(t), t) = 0,

vT (t)[(

1N

1N×N

)⊗ In

]f (x(t − τ), v(t − τ), x(t), v(t), t) = 0.

(12)

Combining (10)-(12), one obtains

V(t) =[αxT (t) + βvT (t)

] [f (x(t − τ), v(t − τ), x(t), v(t), t) − 1N ⊗ f (x(t − τ), v(t − τ), x(t), v(t), t)

]+yT (t)

−α2

2 (L + LT ) ON

ON − β2

2 (L + LT ) + αIN

⊗ In

y(t). (13)

By Assumption 1, one gets

αxT (t)[f (x(t − τ), v(t − τ), x(t), v(t), t) − 1N ⊗ f (x(t − τ), v(t − τ), x(t), v(t), t)

]= α

N∑i=1

(xi(t) − x(t))T [f (xi(t − τ), vi(t − τ), xi(t), vi(t), t) − f (x(t − τ), v(t − τ), x(t), v(t), t)

]≤ α

N∑i=1

∥xi(t)∥ (ρ1∥xi(t − τ)∥ + ρ2∥vi(t − τ)∥ + ρ3∥xi(t)∥ + ρ4∥vi(t)∥)

≤ (ρ1 + ρ2

2+ ρ3)

N∑i=1

∥xi(t)∥2 +ρ1

2

N∑i=1

∥xi(t − τ)∥2 + ρ2

2

N∑i=1

∥vi(t − τ)∥2 + ρ4

N∑i=1

∥xi(t)∥∥vi(t)∥, (14)

7

and

βvT (t)[f (x(t − τ), v(t − τ), x(t), v(t), t) − 1N ⊗ f (x(t − τ), v(t − τ), x(t), v(t), t)

]= β

N∑i=1

(vi(t) − v(t))T [f (xi(t − τ), vi(t − τ), xi(t), vi(t), t) − f (x(t − τ), v(t − τ), x(t), v(t), t)

]≤ β

N∑i=1

∥vi(t)∥ (ρ1∥xi(t − τ)∥ + ρ2∥vi(t − τ)∥ + ρ3∥xi(t)∥ + ρ4∥vi(t)∥)

≤ β

ρ1

2

N∑i=1

∥xi(t − τ)∥2 +(ρ1 + ρ2

2+ ρ4

) N∑i=1

∥vi(t)∥2 +ρ2

2

N∑i=1

∥vi(t − τ)∥2 + ρ3

N∑i=1

∥xi(t)∥∥vi(t)∥ . (15)

Combining (13)-(15) gives

V(t) ≤ (ρ1 + ρ2 + 2ρ3)α2

N∑i=1

∥xi(t)∥2 +(α + β)ρ1

2

N∑i=1

∥xi(t − τ)∥2 + (ρ1 + ρ2 + 2ρ4)β2

N∑i=1

∥vi(t)∥2

+(α + β)ρ2

2

N∑i=1

∥vi(t − τ)∥2 + (βρ3 + αρ4)N∑

i=1

∥xi(t)∥∥vi(t)∥

+ yT (t)

−α2

2 (L + LT ) ON

ON − β2

2 (L + LT ) + αIN

⊗ In

y(t)

≤

(ρ1 + ρ2 + 2ρ3 − αλ2(L + LT )

)α

2

N∑i=1

∥xi(t)∥2 + (βρ3 + αρ4)N∑

i=1

∥xi(t)∥∥vi(t)∥

+(ρ1 + ρ2 + 2ρ4)β + 2α − β2λ2(L + LT )

2

N∑i=1

∥vi(t)∥2 +(α + β)ρ1

2

N∑i=1

∥xi(t − τ)∥2

+(α + β)ρ2

2

N∑i=1

∥vi(t − τ)∥2

=12

(−∥y(t)∥T (R1 ⊗ IN) ∥y(t)∥ + ∥y(t − τ)∥T (S 1 ⊗ IN) ∥y(t − τ)∥

), (16)

where R1 =

(αλ2(L + LT ) − ρ1 − ρ2 − 2ρ3

)α − (βρ3 + αρ4)

− (βρ3 + αρ4) β2λ2(L + LT ) − (ρ1 + ρ2 + 2ρ4)β − 2α

, S 1 =

(α + β)ρ1 0

0 (α + β)ρ2

,∥x(t)∥ = (∥x1(t)∥, · · · , ∥xN(t)∥)T , ∥v(t)∥ = (∥v1(t)∥, · · · , ∥vN(t)∥)T , ∥y(t)∥ = (∥x(t)∥T , ∥v(t)∥T )T , ∥y(t − τ)∥ = (∥x(t −τ)∥T , ∥v(t − τ)∥T )T . On the other hand, one has

V(t) =12

yT (t) (P ⊗ In) y(t),

=αβ

2xT (t)

((L + LT ) ⊗ In

)x(t) + αxT (t)v(t) +

β

2vT (t)v(t)

≤ 12

yT (t) (P1 ⊗ INn) y(t), (17)

8

where P1 =

αβλN(L + LT ) a

a β

. Thus, according to Eq. (16) and the following facts:

V(t) ≤ 12λ2(P1)yT (t)y(t),

V(t − τ) ≥ 12λ1(Q)yT (t − τ)y(t − τ),

∥y(t)∥T R1∥y(t)∥ ≥ λ1(R1)yT (t)y(t),

∥y(t − τ)∥T S 1∥y(t − τ)∥ ≤ λ2(S 1)yT (t − τ)y(t − τ),

one obtains

V(t) ≤ −γ1V(t) + γ2V(t − τ), (18)

where γ1 =λ1(R1)λ2(P1) , γ2 =

c0λ1(Q) , and c0 = (α + β) max{ρ1, ρ2}.

For kω + δ < t < (k + 1)ω, k ∈ ℕ, taking the time derivative of V(t) along the trajectories of (8) gives

V(t) = yT (t)(P ⊗ In)[F(x(t − τ), v(t − τ), x(t), v(t), t) + (B2 ⊗ In)y(t)] (19)

= αxT[(

IN −1N

1N×N

)⊗ In

]f (x(t − τ), v(t − τ), x(t), v(t), t) + βvT

[(IN −

1N

1N×N

)⊗ In

]f (x(t − τ), v(t − τ), x(t), v(t), t) + yT (t) [(B2 ⊗ In)] y(t).

Similar to the previous analysis, one obtains

V(t) ≤ (ρ1 + ρ2 + 2ρ3)α2

N∑i=1

∥xi(t)∥2 +(α + β)ρ1

2

N∑i=1

∥xi(t − τ)∥2 + (ρ1 + ρ2 + 2ρ4)β2

N∑i=1

∥vi(t)∥2

+(α + β)ρ2

2

N∑i=1

∥vi(t − τ)∥2 + (βρ3 + αρ4)N∑

i=1

∥xi(t)∥∥vi(t)∥ + yT (t)

ON IN

ON ON

⊗ In

y(t)

=12

(∥y(t)∥T (R2 ⊗ IN) ∥y(t)∥ + ∥y(t − τ)∥T (S 2 ⊗ IN) ∥y(t − τ)∥

), (20)

where R2 =

(ρ1 + ρ2 + 2ρ3)α βρ3 + αρ4 + 1

βρ3 + αρ4 + 1 (ρ1 + ρ2 + 2ρ4)β

, S 2 =

(α + β)ρ1 0

0 (α + β)ρ2

. It then follows that

V(t) ≤ λ2(R2)λ1(Q)

V(t) +λ2(S 2)λ1(Q)

V(t − τ)

≤ γ3V(t) + γ4V(t − τ), (21)

where γ3 =c1+c2+

√(c1−c2)2+c3

λ1(Q) , γ4 =c0

λ1(Q) , c1 =(ρ1+ρ2+2ρ3)α

2 , c2 =(ρ1+ρ2+2ρ4)β

2 , c3 = (βρ3 + αρ4 + 1)2, c0 = (α +

β) max{ρ1, ρ2}.Based on the above analysis and according to Lemma 4, one obtains

V(t) ≤ |V(0)|τe−rt, 0 ≤ t ≤ δ, (22)

where r is the unique positive solution of −r = −γ1 + γ2erτ, |V(0)|τ = max−τ≤s≤0

V(s). For δ < t < ω, by using Lemma 5,

one obtains

V(t) ≤ |V(δ)|τe(γ3+γ4)t. (23)

9

Then, according to (22), one has

|V(δ)|τ = maxδ−τ≤t≤δ

V(t) ≤ |V(0)|τe−r(δ−τ). (24)

Combining (23) and (24) yields

V(t) ≤ |V(δ)|τe(γ3+γ4)(t−δ) ≤ |V(0)|τe−r(δ−τ)+(γ3+γ4)(t−δ), δ < t < ω. (25)

As V(t) is a continuous function of t, one has

V(ω) = limt→ω−

V(t) ≤ |V(0)|τe−r(δ−τ)+(γ3+γ4)(ω−δ). (26)

Then,

|V(ω)|τ = maxω−τ≤t≤ω

V(t)

≤ |V(δ)|τe(γ3+γ4)(ω−δ)

≤ |V(0)|τe−r(δ−τ)+(γ3+γ4)(ω−δ) = |V(0)|τe−∆, (27)

where ∆ = r(δ − τ) − (γ3 + γ4)(ω − δ) > 0. For any positive integer k, one has

|V(kω)|τ ≤ |V(0)|τe−k∆. (28)

For arbitrary t > 0, there exists a non-negative integer k, such that kω < t ≤ (k + 1)ω. When t ∈ (kω, kω + δ], one

obtains

V(t) ≤ |V(kω)|τe−r(t−kω)

≤ |V(0)|τe−k∆−r(t−kω)

≤ |V(0)|τe−k∆

≤ |V(0)|τe∆e−(∆ω )t. (29)

When t ∈ (kω + δ, (k + 1)ω], one has

V(t) ≤ |V(kω + δ)|τe(γ3+γ4)(t−kω−δ)

≤ |V(0)|τe−k∆−rδe(γ3+γ4)(ω−δ)

≤ |V(0)|τe(γ3+γ4)(ω−δ)−rδ+∆e−(∆ω )t

= |V(0)|τe−rτe−(∆ω )t. (30)

Combining (29)-(30) gives

V(t) ≤ K0e−(∆ω )t, for all t > 0, (31)

where K0 = e∆|V(0)|τ, which indicates that the states of agents exponentially converge to consensus. This completes

the proof.

Corollary 1. Suppose that the communication topology G(A) is a strongly connected and balanced network, and

Assumption 1 holds. Then, second-order consensus in system (5) is achieved if the following conditions hold:

10

(i) β > α,

(ii) λ2(L + LT ) > max{α−1, ϱ1, ϱ2

},

(iii) δ > rτ+(γ3+γ4)ωr+γ3+γ4

,

where ϱ1 =α(ρ1+ρ2+2ρ3)+βρ3+αρ4

α2 +max{ρ1,ρ2}(α+β)(βλN (L+LT )+1)

α(β−α) , ϱ2 =β(ρ1+ρ2+2ρ4)+2α

β2 +max{ρ1,ρ2}(α+β)[αβλN (L+LT )+α]

β2(β−α) , r is the

unique positive solution of −r = −γ1 + γ2erτ, γ1 =min{κ1,κ2}

αβλN (L+LT )+α , γ2 =(α+β) max{ρ1,ρ2}

β−α , γ3 =c1+c2+

√(c1−c2)2+c3

β−α ,

γ4 =(α+β) max{ρ1,ρ2}

β−α , κ1 = β2λ2(L + LT ) − β(ρ1 + ρ2 + 2ρ3) − (βρ3 + αρ4), κ2 = α

2λ2(L + LT ) − α(ρ1 + ρ2 + 2ρ4) − 2α,

c1 =(ρ1+ρ2+2ρ3)α

2 , c2 =(ρ1+ρ2+2ρ4)β

2 , and c3 = (βρ3 + αρ4 + 1)2.

Proof: Construct the same Lyapunov function candidate V(t) as that in the proof of Theorem 1. By the Gersgorin

disk theorem [43] and conditions (i) and (ii), the Corollary can be proved by following the proof of Theorem 1.

Remark 2. In Ref. [37], the concept of general algebraic connectivity a(L) is introduced to describe the second-

order multi-agent system’s ability to reach consensus. By Definition 6 in Ref. [37], one has a(L) = λ2(L+LT )2 for a

strongly connected and balanced G(A), where L is the Laplacian matrix of the graph. Suppose that β > α. Then,

from the Corollary 1, the second-order consensus can be achieved if the general algebraic connectivity a(L) and the

communication time duration δ are larger than their corresponding threshold values, respectively.

In practice, the communication topology among agents may not be fixed because of the restrictions of physical

equipments or the signal interference. Therefore, it is more reasonable to assume that the communication topology

is dynamically switching. Let G = {G(A1), · · · ,G(Aπ)} be a set of possible topologies. For convenience, introduce a

switching signal σ : [0,∞) → Π, where Π = {1, · · · , π}. Denote by Lσ(t) the Laplacian matrix of G(Aσ(t)). Then, the

following theorem and corollary can be obtained, for which the proofs are straight forward therefore omitted.

Theorem 2. Suppose that the communication topology G(Aσ(t)) is kept strongly connected and balanced through-

out the process and, moreover, Assumption 1 holds. Then, second-order consensus in system (5) is achieved if the

following conditions hold:

(i) λ2(Li + LTi ) > α2

β,

(ii) λ1(Ri1) > c0λ2(Pi

1)λ1(Qi) ,

(iii) δ > riτ+(γi3+γ

i4)ω

ri+γi3+γ

i4

,

where Ri1 =

(αλ2(Li + LT

i ) − ρ1 − ρ2 − 2ρ3

)α − (βρ3 + αρ4)

− (βρ3 + αρ4) β2λ2(Li + LTi ) − (ρ1 + ρ2 + 2ρ4)β − 2α

, Pi1 =

αβλmax(Li + LTi ) α

α β

,Qi =

αβλ2(Li + LTi ) α

α β

, c0 = (α + β) max{ρ1, ρ2}, and ri is the unique positive solution of −ri = −γi1 + γ

i2eriτ,

γi1 =

λ1(Ri1)

λ2(Pi1) , γ2 =

c0λ1(Qi) , γ

i3 =

c1+c2+√

(c1−c2)2+c3

λ1(Qi) , γi4 =

c0λ1(Qi) , c1 =

(ρ1+ρ2+2ρ3)α2 , c2 =

(ρ1+ρ2+2ρ4)β2 , c3 = (βρ3 + αρ4 +

1)2, i ∈ Π.

Corollary 2. Suppose that the communication topology G(Aσ(t)) is kept strongly connected and balanced through-

out the process and, moreover, Assumption 1 holds. Then, second-order consensus in system (5) is achieved if the

following conditions hold:

(i) β > α,

(ii) mini∈Π λ2(Li + LTi ) > max

{α−1, ϱ0

},

(iii) δ > rτ+(γ3+γ4)ωr+γ3+γ4

,

where ϱ0 = maxi∈Π

{ϱi

1, ϱi2

}, ϱi

1 =α(ρ1+ρ2+2ρ3)+βρ3+αρ4

α2 +ρ0(α+β)(βλN (Li+LT

i )+1)α(β−α) , ϱi

2 =β(ρ1+ρ2+2ρ4)+2α

β2 +ρ0(α+β)[αβλN (Li+LT

i )+α]β2(β−α) ,

11

r is the unique positive solution of −r = −γ1 + γ2erτ, γ1 =κmin

αβλN (L+LT )+α , γ2 =(α+β) max{ρ1,ρ2}

β−α , γ3 =c1+c2+

√(c1−c2)2+c3

β−α ,

γ4 =(α+β) max{ρ1,ρ2}

β−α , κmin = mini∈Π{κi

1, κi2}, κi

1 = β2λ2(Li + LT

i ) − β(ρ1 + ρ2 + 2ρ3) − (βρ3 + αρ4), κi2 = α

2λ2(Li + LTi ) −

α(ρ1 + ρ2 + 2ρ4) − 2α, c1 =(ρ1+ρ2+2ρ3)α

2 , c2 =(ρ1+ρ2+2ρ4)β

2 , c3 = (βρ3 + αρ4 + 1)2, i ∈ Π.

4. A Simulation Example

In this section, a simulation example is provided to verify the theoretical analysis.

Consider the second-order consensus protocol with time-delayed nonlinear velocities in system (5), where the

communication topology is shown in Fig. 1 with weighting on the edges. The time-delayed nonlinear function f is

described by time-delayed Chua’s circuit [44]:

f (xi(t − τ), vi(t − τ), xi(t), vi(t), t) =

µ (−vi1 + vi2 − l(vi1))

vi1 − vi2 + vi3

−ςvi2 − ϵsin(σvi1(t − τ))

, i = 1, · · · , 4, (32)

where l(vi1) = bvi1 + 0.5(a − b)(|vi1 + 1| − |vi1 − 1|), xi = [xi1, xi2, xi3]T , vi = [vi1, vi2, vi3]T . The isolated system

(32) is chaotic when µ = 10, ς = 18, ϵ = 0.02, σ = 0.02, τ = 0.01, a = −4/3 and b = −3/4, as shown in

Fig. 1 with initial conditions vi(t) = [0.016, 0.018,−0.015]T , t ∈ [−τ, 0]. In view of Assumption 1, one obtains

ρ1 = 0, ρ2 = 0.0004, ρ3 = 0, ρ4 = 4.3871. Let α = 11.5, β = 12, δ = 0.485, and ω = 0.5. From Fig. 2,

it is easy to see that the communication topology G(A) is strongly connected and balanced. By Theorem 1, one

has λ2(L + LT ) = 12 > α2

β= 11.0208, λ1(R1) = 1555.7 > c0λ2(P1)

λ1(Q) = 3.0473, and δ = 0.485 > rτ+(γ3+γ4)ωr+γ3+γ4

=

0.4845. Therefore, second-order consensus can be achieved in multi-agent system (5). The position and velocity

states of all agents are shown in Fig. 2, with initial conditions x1(t) = [0.25,−0.13, 0.04]T , x2(t) = [2, 1.5, 2.5]T ,

x3(t) = [−1,−1.5,−2.5]T , x4(t) = [−2,−0.8, 0.3]T , v1(t) = [3.016, 2.018, 0.085]T , v2(t) = [2.016, 3.018, 1.085]T ,

v3(t) = [−1.085,−1.282, 1.285]T , v4(t) = [−2.085,−0.582,−0.015]T , for t ∈ [−τ, 0]. Simulation results shown in

Figs. (3) and (4) verify the theoretical analysis very well.

−4 −2 0 2 4−1

−0.5

0

0.5

1−10

−5

0

5

10

vi1

vi2

v i3

Figure 1: Chaotic trajectory of the model (32).

12

42

1

3

6 6

4

6 6

4

Figure 2: Communication topology G(A).

0 1 2 3 4 5 6 7−15

−10

−5

0

5

10

15

t

x

Figure 3: Consensus of state trajectories of multiple agents.

5. Conclusions

In this paper, a novel second-order intermittent consensus protocol for multi-agent systems with time-delayed

nonlinear dynamics and switching communication topologies has been introduced and studied. It has been shown that

second-order consensus can be reached if the communication time duration and the general algebraic connectivity

are larger than their corresponding thresholds, respectively. Future work will be focused on the consensus behaviors

of more complicated and practical models, such as second-order multi-agent systems with nonlinear dynamics and

transmission delays, higher-order multi-agent systems with nonlinear dynamics, and so on.

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